Surgery and the Spinorial τ-Invariant

Abstract

We associate to a compact spin manifold M a real-valued invariant τ (M ) by taking the supremum over all conformal classes of the infimum inside each conformal class of the first positive Dirac eigenvalue, when the metrics are normalized to unit volume. This invariant is a spinorial analogue of Schoen's σ-constant, also known as the smooth Yamabe invariant.We prove that if N is obtained from M by surgery of codimension at least 2 then τ (N ) ≥ min{τ (M ), Λn}, where Λn is a positive constant depending only on n = dim M . Various topological conclusions can be drawn, in particular that τ is a spin-bordism invariant below Λn. Also, below Λn the values of τ cannot accumulate from above when varied over all manifolds of dimension n.

Smooth Yamabe invariant and surgery

“…At the end of the section we want to mention some similar constructions in the literature. An analogous surgery formula holds if we replace the Conformal Laplacian by the Dirac operator, see [4] for details and applications. D. Joyce [28], followed by L. Mazzieri [42,43], considered a problem tightly related to our result: their goal is to construct a metric on a manifold obtained via a connected sum along a k-dimensional submanifold.…”

Relations Between Threshold Constants for Yamabe Type Bordism Invariants

“…In the articles [5] and [4] the following situation was considered. Assume that N m is obtained from M m by a surgery of dimension k. Then for any metric g on M a family of special metrics g ϑ , ϑ > 0, was constructed.…”